In algebra, Hall's universal group is a countable locally finite group, say U, which is uniquely characterized by the following properties.
It was defined by Philip Hall in 1959, and has the universal property that all countable locally finite groups embed into it.
Construction
Take any group
and so on. Since a group acts faithfully on itself by permutations
according to Cayley's theorem, this gives a chain of monomorphisms
A direct limit (that is, a union) of all
Indeed, U then contains a symmetric group of arbitrarily large order, and any group admits a monomorphism to a group of permutations, as explained above. Let G be a finite group admitting two embeddings to U. Since U is a direct limit and G is finite, the images of these two embeddings belong to